Functions and Their Graphs

SECTION
3.4

Operations on Functions and Composition of Functions

SKILLS OBJECTIVES


■��	Add, subtract, multiply, and divide functions.

■��	Evaluate composite functions.

■��	Determine the domain of functions resulting from operations on and composition of functions.

CONCEPTUAL OBJECTIVES


■��	Understand domain restrictions when dividing functions.

■��	Realize that the domain of a composition of functions excludes values that are not in the domain of the inside function.

�

Two different functions can be combined using mathematical operations such as addition, subtraction, multiplication, and division. Also, there is an operation on functions called composition, which can be thought of as a function of a function. When we combine functions, we do so algebraically. Special attention must be paid to the domain and range of the combined functions.

Adding, Subtracting, Multiplying, and Dividing Functions

Consider the two functions

and

. The domain of both of these functions is the set of all real numbers. Therefore, we can add, subtract, or multiply these functions for any real number

The result is in fact a new function, which we denote:

Although both

and

are defined for all real numbers

, we must restrict

so that

to form the quotient

The result is in fact a new function, which we denote:

Two functions can be added, subtracted, and multiplied. The resulting function domain is therefore the intersection of the domains of the two functions. However, for division, any value of

(input) that makes the denominator equal to zero must be eliminated from the domain.

The previous examples involved polynomials. The domain of any polynomial is the set of all real numbers. Adding, subtracting, and multiplying polynomials result in other polynomials, which have domains of all real numbers. Let's now investigate operations applied to functions that have a restricted domain.

The domain of the sum function, difference function, or product function is the intersection of the individual domains of the two functions. The quotient function has a similar domain in that it is the intersection of the two domains. However, any values that make the denominator zero must also be eliminated.

Function	Notation	Domain
Sum
Difference
Product
Quotient

We can think of this in the following way: Any number that is in the domain of both the functions is in the domain of the combined function. The exception to this is the quotient function, which also eliminates values that make the denominator equal to zero.

EXAMPLE�1�

Operations on Functions: Determining Domains of New Functions

For the functions

and

, determine the sum function, difference function, product function, and quotient function. State the domain of these four new functions.

Solution

Sum function:
Difference function:
Product function:
Quotient function:

The domain of the square root function is determined by setting the argument under the radical greater than or equal to zero.

The domain of the sum, difference, and product functions is

The quotient function has the additional constraint that the denominator cannot be zero. This implies that

, so the domain of the quotient function is

■	YOUR TURN�

Given the function

and

, find

and state its domain.

�EXAMPLE�2�

Quotient Function and Domain Restrictions

Given the functions

and

, find the quotient function,

, and state its domain.

Solution

The quotient function is written as

Domain of

The real numbers that are in both the domain for

and the domain for

are represented by the intersection

. Also, the denominator of the quotient function is equal to zero when

, so we must eliminate this value from the domain.

■	YOUR TURN�

For the functions given in Example 2, determine the quotient function

, and state its domain.

Composition of Functions

Recall that a function maps every element in the domain to exactly one corresponding element in the range as shown in the figure below.

Suppose there is a sales rack of clothes in a department store. Let

correspond to the original price of each item on the rack. These clothes have recently been marked down

. Therefore, the function

represents the current sale price of each item. You have been invited to a special sale that lets you take

off the current sale price and an additional

off every item at checkout. The function

determines the checkout price. Note that the output of the function

is the input of the function

as shown in the figure below.

This is an example of a composition of functions, when the output of one function is the input of another function. It is commonly referred to as a function of a function.

An algebraic example of this is the function

. Suppose we let

and

. Recall that the independent variable in function notation is a placeholder. Since

, then

. Substituting the expression for

, we find

. The function

is said to be a composite function,

Note that the domain of

is the set of all real numbers, and the domain of

is the set of all nonnegative numbers. The domain of a composite function is the set of all

such that

is in the domain of

. For instance, in the composite function

, we know that the allowable inputs into

are all numbers greater than or equal to zero. Therefore, we restrict the outputs of

and find the corresponding

. Those

are the only allowable inputs and constitute the domain of the composite function

The symbol that represents composition of functions is a small open circle; thus

and is read aloud as “

.” It is important not to confuse this with the multiplication sign:

CAUTION�

COMPOSITION OF FUNCTIONS

Given two functions

and

, there are two composite functions that can be formed.

Notation	Words	Definition	Domain
	composed with		The set of all real numbers in the domain of such that is also in the domain of .
	composed with		The set of all real numbers in the domain of such that is also in the domain of .

It is important to realize that there are two “filters” that allow certain values of

into the domain. The first filter is

. If

is not in the domain of

, it cannot be in the domain of

. Of those values for

that are in the domain of

, only some pass through, because we restrict the output of

to values that are allowable as input into

. This adds an additional filter.

The domain of

is always a subset of the domain of

, and the range of

is always a subset of the range of

�EXAMPLE�3�

Finding a Composite Function

Given the functions

and

, find

Solution

Write using placeholder notation.
Express the composite function .
Substitute into .
Eliminate the parentheses on the right side.

■	YOUR TURN�

Given the functions in Example 3, find

�EXAMPLE�4�

Determining the Domain of a Composite Function

Given the functions

and

, determine

, and state its domain.

Solution

Write using placeholder notation.
Express the composite function .
Substitute into .
Multiply the right side by .

What is the domain of

? By inspecting the final result of

, we see that the denominator is zero when

. Therefore,

. Are there any other values for

that are not allowed? The function

has the domain

; therefore we must also exclude zero.

The domain of

excludes

and

or, in interval notation,

■	YOUR TURN�

For the functions

and

given in Example 4, determine the composite function

and state its domain.

The domain of the composite function cannot always be determined by examining the final form of

CAUTION�

The domain of the composite function cannot always be determined by examining the final form of

EXAMPLE�5�

Determining the Domain of a Composite Function (Without Finding the Composite Function)

Let

and

. Find the domain of

. Do not find the composite function.

Solution

Find the domain of .
Find the range of .

, the output of

becomes the input for

. Since the domain of

is the set of all real numbers except 2, we eliminate any values of

in the domain of

that correspond to

Let .
Square both sides.
Solve for .

Eliminate

from the domain of

State the domain of

�EXAMPLE�6�

Evaluating a Composite Function

Given the functions

and

, evaluate:

(a)��

(b)��

(c)��

(d)��

Solution

One way of evaluating these composite functions is to calculate the two individual composites in terms of

and

. Once those functions are known, the values can be substituted for

and evaluated.

Another way of proceeding is as follows:

(a)��

Write the desired quantity.
Find the value of the inner function .
Substitute into .
Evaluate .

(b)��

Write the desired quantity.
Find the value of the inner function .
Substitute into .
Evaluate .

(c)��

Write the desired quantity.
Find the value of the inner function .
Substitute into .
Evaluate .

(d)��

Write the desired quantity.
Find the value of the inner function .
Substitute into .
Evaluate .

■	YOUR TURN�

Given the functions

and

, evaluate

and

3.4.2.1�Application Problems

Recall the example at the beginning of this chapter regarding the clothes that are on sale. Often, real-world applications are modeled with composite functions. In the clothes example,

is the original price of each item. The first function maps its input (original price) to an output (sale price). The second function maps its input (sale price) to an output (checkout price). Example 7 is another real-world application of composite functions.

Three temperature scales are commonly used:

■��

The degree Celsius

scale


●��	This scale was devised by dividing the range between the freezing and boiling points of pure water at sea level into 100 equal parts. This scale is used in science and is one of the standards of the “metric” (SI) system of measurements.

■��

The Kelvin (K) temperature scale


●��	This scale shifts the Celsius scale down so that the zero point is equal to absolute zero (about ), a hypothetical temperature at which there is a complete absence of heat energy.

●��	Temperatures on this scale are called kelvins, not degrees kelvin, and kelvin is not capitalized. The symbol for the kelvin is K.

■��

The degree Fahrenheit

scale


●��	This scale evolved over time and is still widely used mainly in the United States, although Celsius is the preferred “metric” scale.

●��	With respect to pure water at sea level, the degrees Fahrenheit are gauged by the spread from (freezing) to (boiling).

The equations that relate these temperature scales are

EXAMPLE�7�

Applications Involving Composite Functions

Determine degrees Fahrenheit as a function of kelvins.

Solution

Degrees Fahrenheit is a function of degrees Celsius.
Now substitute into the equation for .
Simplify.
�

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SECTION
3.4

SUMMARY

Operations on Functions

Function	Notation
Sum
Difference
Product
Quotient

The domain of the sum, difference, and product functions is the intersection of the domains, or common domain shared by both

and

. The domain of the quotient function is also the intersection of the domain shared by both

and

with an additional restriction that

Composition of Functions

The domain restrictions cannot always be determined simply by inspecting the final form of

. Rather, the domain of the composite function is a subset of the domain of

. Values of

must be eliminated if their corresponding values of

are not in the domain of

SECTION
3.4

EXERCISES

SKILLS

In Exercises 1-10, given the functions

and

, find

, and

, and state the domain of each.

1.��

2.��

3.��

4.��

5.��

6.��

7.��

8.��

9.��	Domains: Must have both and . So, . For the quotient, must have both and . So, .

10.��

In Exercises 11-20, for the given functions

and

, find the composite functions

and

, and state their domains.

11.��

12.��

13.��

14.��

15.��

16.��

17.��

18.��

19.��

20.��

In Exercises 21-38, evaluate the functions for the specified values, if possible.

21.��

22.��

23.��

24.��

25.��

26.��

27.��

28.��

29.��

30.��

31.��

32.��

33.��

34.��

35.��

36.��

37.��

38.��

In Exercises 39-50, evaluate

and

, if possible.

39.��

40.��

41.��

42.��

43.��

44.��

45.��

46.��

47.��

48.��

49.��

50.��

In Exercises 51-60, show that

and

51.��

52.��

53.��

54.��

55.��

56.��

57.��

58.��

59.��

60.��

In Exercises 61-66, write the function as a composite of two functions

and

. (More than one answer is correct.)

61.��

62.��

63.��

64.��

65.��

66.��

APPLICATIONS

Exercises 67 and 68 depend on the relationship between degrees Fahrenheit, degrees Celsius, and kelvins:

67.��

Temperature.

Write a composite function that converts kelvins into degrees Fahrenheit.

68.��

Temperature.

Convert the following degrees Fahrenheit to kelvins:

and

69.��

Dog Run.

Suppose that you want to build a square fenced-in area for your dog. Fencing is purchased in linear feet.

(a)��

Write a composite function that determines the area of your dog pen as a function of how many linear feet are purchased.

(b)��

If you purchase 100 linear feet, what is the area of your dog pen?

(c)��

If you purchase 200 linear feet, what is the area of your dog pen?

70.��

Dog Run.

Suppose that you want to build a circular fenced-in area for your dog. Fencing is purchased in linear feet.

(a)��

Write a composite function that determines the area of your dog pen as a function of how many linear feet are purchased.

(b)��

If you purchase 100 linear feet, what is the area of your dog pen?

(c)��

If you purchase 200 linear feet, what is the area of your dog pen?

71.��

Market Price.

Typical supply and demand relationships state that as the number of units for sale increases, the market price decreases. Assume that the market price

and the number of units for sale

are related by the demand equation:

Assume that the cost

of producing

items is governed by the equation

and the revenue

generated by selling

units is governed by

(a)��

Write the cost as a function of price

(b)��

Write the revenue as a function of price

(c)��

Write the profit as a function of price

72.��

Market Price.

Typical supply and demand relationships state that as the number of units for sale increases, the market price decreases. Assume that the market price

and the number of units for sale

are related by the demand equation:

Assume that the cost

of producing

items is governed by the equation

and the revenue

generated by selling

units is governed by

(a)��

Write the cost as a function of price

(b)��

Write the revenue as a function of price

(c)��

Write the profit as a function of price

In Exercises 73 and 74, refer to the following:

The cost of manufacturing a product is a function of the number of hours

the assembly line is running per day. The number of products manufactured

is a function of the number of hours

the assembly line is operating and is given by the function

. The cost of manufacturing the product

measured in thousands of dollars is a function of the quantity manufactured, that is, the function

73.��

Business.

If the quantity of a product manufactured during a day is given by

and the cost of manufacturing the product is given by

(a)��

Find a function that gives the cost of manufacturing the product in terms of the number of hours

the assembly line was functioning,

(b)��

Find the cost of production on a day when the assembly line was running for 16 hours. Interpret your answer.

74.��

Business.

If the quantity of a product manufactured during a day is given by

and the cost of manufacturing the product is given by

(a)��

Find a function that gives the cost of manufacturing the product in terms of the number of hours

the assembly line was functioning,

(b)��

Find the cost of production on a day when the assembly line was running for 24 hours. Interpret your answer.

In Exercises 75 and 76, refer to the following:

Surveys performed immediately following an accidental oil spill at sea indicate the oil moved outward from the source of the spill in a nearly circular pattern. The radius of the oil spill

measured in miles is a function of time

measured in days from the start of the spill, while the area of the oil spill is a function of radius, that is, the function

75.��

Environment: Oil Spill.

If the radius of the oil spill is given by

and the area of the oil spill is given by

(a)��

Find a function that gives the area of the oil spill in terms of the number of days since the start of the spill,

(b)��

Find the area of the oil spill to the nearest square mile 7 days after the start of the spill.

76.��

Environment: Oil Spill.

If the radius of the oil spill is given by

and the area of the oil spill is given by

(a)��

Find a function that gives the area of the oil spill in terms of the number of days since the start of the spill,

(b)��

Find the area of the oil spill to the nearest square mile 5 days after the start of the spill.

77.��

Environment: Oil Spill.

An oil spill makes a circular pattern around a ship such that the radius in feet grows as a function of time in hours

. Find the area of the spill as a function of time.

78.��

Pool Volume.

A 20 foot by 10 foot rectangular pool has been built. If 50 cubic feet of water is pumped into the pool per hour, write the water-level height (feet) as a function of time (hours).

79.��

Fireworks.

A family is watching a fireworks display. If the family is 2 miles from where the fireworks are being launched and the fireworks travel vertically, what is the distance between the family and the fireworks as a function of height above ground?

80.��

Real Estate.

A couple are about to put their house up for sale. They bought the house for

a few years ago, and when they list it with a realtor they will pay a

commission. Write a function that represents the amount of money they will make on their home as a function of the asking price

CATCH THE MISTAKE

In Exercises 81-86, for the functions

and

, find the indicated function and state its domain. Explain the mistake that is made in each problem.

81.��

Solution:

Domain:

This is incorrect. What mistake was made?

82.��

Solution:

Domain:

This is incorrect. What mistake was made?

83.��

Solution:

Domain:

This is incorrect. What mistake was made?

84.��

Given the function

and

, find

, and state the domain.

Solution:

Domain:

This is incorrect. What mistake was made?

85.��

Domain:

This is incorrect. What mistake was made?

86.��

Domain:

This is incorrect. What mistake was made?

CONCEPTUAL

In Exercises 87-90, determine whether each statement is true or false.

87.��

When adding, subtracting, multiplying, or dividing two functions, the domain of the resulting function is the union of the domains of the individual functions.

88.��

For any functions

and